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DISPERSIVE ORDERING FOR THE MULTIVARIATE NORMAL DISTRIBUTION

Published online by Cambridge University Press:  05 January 2016

Xuan Leng ,
Jinsen Zhuang  and
Taizhong Hu
Xuan Leng
Affiliation:
Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei, Anhui 230026, China E-mail: [email protected]
Jinsen Zhuang
Affiliation:
Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei, Anhui 230026, China E-mail: [email protected]
Taizhong Hu
Affiliation:
Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei, Anhui 230026, China E-mail: [email protected]
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Abstract

Let (X1, …, Xn) be a multivariate normal random vector with any mean vector, variances equal to 1 and covariances equal and positive. Turner and Whitehead [9] established that the largest order statistic max{X1, …, Xn} is less than the standard normal random variable in the dispersive order. In this paper, we give a new and straightforward proof for this result. Several possible extensions of this result are also discussed.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 30 , Issue 2 , April 2016 , pp. 141 - 152
Copyright
Copyright © Cambridge University Press 2016 

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References

1.Bickel & Lehmann (1976). Descriptive statistics for non-parametric models. III. Dispersion. Annals of Statistics 4: 1139–1158.Google Scholar
2.Jeon, J., Kochar, S., & Park, C.G. (2006). Dispersive ordering – some applications and examples. Statistical Papers 47: 227–247.CrossRefGoogle Scholar
3.Khaledi, B.-E. & Kochar, S. (2000). On dispersive ordering between order statistics in one-sample and two-sample problems. Statistics and Probability Letters 46: 257–261.Google Scholar
4.Lewis, T. & Thompson, J.W. (1981). Dispersive distributions and the connection between dispersivity and strong unimodality. Journal of Applied Probability 18: 76–90.CrossRefGoogle Scholar
5.Sampford, M.R. (1953). Some inequalities on Mill's ratio and related functions. Annals of Mathematical Statistics 24: 130–132.Google Scholar
6.Shaked, M. (1982). Dispersive ordering of distributions. Journal of Applied Probability 19: 310–320.Google Scholar
7.Shaked, M. & Shanthikumar, J.G. (2007). Stochastic Orders. New York: Springer.Google Scholar
8.Tong, Y.L. (1990). The Multivariate Normal Distribution. New York: Springer-Verlag.Google Scholar
9.Turner, A. & Whitehead, J. (2015). Partial stochastic dominance for the multivariate Gaussian distribution. Statistics and Probability Letters 103: 80–85.Google Scholar
10.Whitehead, J., Cleary, F., & Turner, A. (2015). Bayesian sample sizes for exploratory clinical trials comparing multiple experimental treatments with a control. Statistics in Medicine DOI: 10.1002/sim.6469.Google Scholar